The built-in newsletter subscription webpart does not have a redirect ability. You can copy the ascx file (CMSWebParts/Newsletters/NewsletterSubscriptionWEbPart.asx) and create a new web part with the functionality you need. You would need to add your redirect logic to the btnSubmit_Click method after the subscriber is saved to the system.
I would also add a property to the new web part for the redirection URL. That way you can set it when the webpart is added to a page and change it later quickly, if needed.
The document has a good guide for cloning web parts here.